In his recent paper, "Restricted nominalism about number and its problems," Stewart Shapiro (Ohio State University), a past fellow of the IIAS and member of the “Computability: Historical, Logical, and Philosophical Foundations” research group, , along with colleagues Eric Snyder and Richard Samuels, delves into a critical analysis of Thomas Hofweber's thesis of "internalism" concerning discourse on natural numbers.

Hofweber's stance, as outlined in his work "Ontology and the Ambitions of Metaphysics," posits that expressions purportedly referring to natural numbers do not actually refer to anything real, and quantification over natural numbers does not involve quantification over genuine objects. This position, termed "internalism," leads Hofweber to establish what he calls "restricted nominalism" about natural numbers, asserting their non-existence while leaving open the possibility of other kinds of abstract entities.

However, Shapiro and his colleagues challenge the efficacy of Hofweber's internalism in establishing restricted nominalism. They argue that Hofweber's primary argument for restricted nominalism is flawed, lacking validity. Moreover, they suggest that Hofweber's proposed analysis of quantification risks collapsing internalism into either a form of traditional error theory or realism, undermining the very foundation of his position.

Click here for the link to the full paper: https://link.springer.com/article/10.1007/s11229-024-04564-5